An efficient numerical algorithm for solving linear systems with cyclic tridiagonal coefficient matrices

Abstract

In the present paper, we mainly consider the direct solution of cyclic tridiagonal linear systems. By using the specific low-rank and Toeplitz-like structure, we derive a structure-preserving factorization of the coefficient matrix. Based on the combination of such matrix factorization and Sherman–Morrison–Woodbury formula, we then propose a cost-efficient algorithm for numerically solving cyclic tridiagonal linear systems, which requires less memory storage and data transmission. Furthermore, we show that the structure-preserving matrix factorization can provide us with an explicit formula for n-th order cyclic tridiagonal determinants. Numerical examples are given to demonstrate the performance and efficiency of our algorithm. All of the experiments are performed on a computer with the aid of programs written in MATLAB.